On Uncertainty Quantification in ParticleAccelerator Modelling

Andreas Adelmanna

aPSI, Switzerland

Abstract

Using a cyclotron based model problem, we demonstrate for the first timethe applicability and usefulness of an uncertainty quantification (UQ) ap-proach in order to construct surrogate models. These surrogate modelquantities for example emittance, energy spread, the halo parameter, andcan be used to construct a global sensitivity model along with error prop-agation and error analysis. The model problem is chosen such that it rep-resents a template for general high intensity particle accelerator modellingtasks. The presented physics problem has to be seen as hypothetical, withthe aim at demonstrating the usefulness and applicability of the presentedUQ approach and not solving a particular problem.

The proposed UQ approach is based on polynomial chaos expansionsand relies on a small number of high fidelity particle accelerator simu-lations. Important uncertainty sources are identified using Sobol’ indiceswithin the global sensitivity analysis.

Keywords: Particle Accelerators, Uncertainty quantification; Polynomialchaos expansion; Global sensitivity analysis

1. INTRODUCTION

Uncertainty Quantification (UQ) describes the origin, propagation, andinterplay of different sources of uncertainties in the analysis and behaviouralprediction of generally complex and high dimensional systems, such asparticle accelerators. With uncertainty, one might question how accuratelya mathematical model can describe the true physics and what impact

Email address: andreas.adelmann@psi.ch (Andreas Adelmann)

Preprint submitted to Journal of Uncertainty Quantification July 17, 2019

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the model uncertainty (structural or parametric) has on the outputs fromthe model. Given a mathematical model, we need to estimate the error.“How accurately is a specified output approximated by a given numericalmethod”? Can the error in the numerical solutions and the specified out-puts be reliably estimated and controlled by adapting resources? For ex-ample, in beam dynamics simulations with space charge, grid sizes wouldbe such a resource.

UQ techniques allow one to quantify output variability in the presenceof uncertainty. These techniques can generally tackle all sources of un-certainties, including structural ones. However, in this paper we focus onparametric uncertainty of input parameters. The moments of the outputdistributions are sampled using Monte Carlo [1] or Quasi-Monte Carlo [2]methods, or newer approaches such as and Multi-Level Monte Carlo [3].Other approaches exist and are known as non-sampling based methods.For an introduction to response surface methods see [4, 5]. The mostpopular method these days, which is used in this paper, is the PolynomialChaos (PC) based method [6]. Strictly speaking, PC also requires sam-pling, but it is not random sampling as in Monte-Carlo type approaches.

Polynomial Chaos (PC) based techniques for propagating uncertaintyand model reduction have been used in the past in almost all importantscientific areas. An incomplete list consists of: climate modelling [7], trans-port in heterogeneous media [8], Ising models [9], combustion [10], fluidflow [11, 12], materials models [13], battery design [14], and Hamiltoniansystems [15].

In probabilistic UQ approaches, one represents uncertain model pa-rameters as random variables or processes. Among these methods, stochas-tic spectral methods [16, 17] based on polynomial chaos (PC) expansions[6, 18] have received special attention due to their advantages over tra-ditional UQ techniques. For a more detailed discussion on that subject,consult the introduction of Hadigol et.al. [14], or alternatively, the book ofSmith [19].

In the field of particle accelerator science, non-intrusive methods arefar more attractive than intrusive methods. The complexity of the physicsmodel would most likely require a total rewrite of the existing simulationpackages, in order to facilitate intrusive methods. Because non-intrusivemethods allow the use of existing beam dynamics codes as black boxes,they are the method of choice. In this paper, we use OPAL as the black-box solver. As we will see later, only independent solution realisations are

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needed, hence embarrassingly parallel implementation is straightforward.The proposed PC approach, first introduced in [16, 20], compute the

statistics for Quantity of Interest (QoI) with a small number of acceleratorsimulations. However in contrast to [16, 20] we do not exploit the sparsityof expansion coefficients, this is subject to further research. Additionally,the presented UQ framework enables one to perform a global sensitivityanalysis (SA) to identify the most important uncertain parameters affectingthe variability of the output quantities.

To avoid confusion, we firstly point out a misnomer by mentioning thatpolynomial chaos [6] and chaos theory [21] are unrelated areas. Origi-nally proposed by Norbert Wiener [6] in 1938 (prior to the development ofchaos theory—hence the unfortunate usage of the term chaos), polyno-mial chaos expansions are a popular method for propagating uncertaintythrough low dimensional systems with smooth dynamics.

This work presents a sampling-based PC approach to study the effectsof uncertainty in various model parameters of accelerators. As a modelproblem, we use the central region of a “PSI Injector 2 like” high inten-sity cyclotron. This paper’s focus is mainly to introduce UQ to the field ofparticle accelerator science and not to solve a particular problem. Withoutloosing generality, we only consider the first 10 turns of the cyclotron.

In Section 2 we present our stochastic modelling approach which isbased on non-intrusive PC expansions. After the derivation of the surro-gate model, we then continue with reviewing a global sensitivity analysisapproach using Sobol’ indices. Section 3 introduces the simulation modeland the model problem. Section 4 applies the UQ to the stated problem,and shows the main features of this approach. The features are generalin nature and not restricted to cyclotrons. Conclusions will be presented inSection 5.

2. UQ VIA POLYNOMIAL CHAOS EXPANSION

Wiener in 1938 [6] introduced polynomial chaos expansion. In 1991,Ghanem and Spanos [16] reintroduced this technique to the field of engi-neering. They first studied problems with Gaussian input uncertainties andextended their method to non-Gaussian random inputs. In their studies, or-thogonal polynomials of the Askey scheme were used. This is known as ageneralised polynomial chaos (gPC) expansion [20]. The method of gPC

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expansion provides a framework to approximate the solution of a stochas-tic system by projecting it onto a basis of polynomials of the random inputs.

An overview and some details on the correspondence between distri-butions and polynomials can be found in [22]. A framework to generatepolynomials for arbitrary distributions has been developed in [23]. The ad-vantage of using polynomial chaos is that it provides exponential conver-gence for smooth models. However, the approach suffers from the curseof dimensionality, making them challenging for problems with number ofparameters in the range 10 . . . 50. To mitigate the curse of dimensionality,sparse grid techniques have traditionally been used [24, 25]. More recently,iterative methods to propagate uncertainty in complex networks have alsobeen developed [26, 27, 28].

2.1. The surrogate modelSuppose you are designing or optimising complex particle accelera-

tors. As a particular example, consider the case of a high intensity hadronmachine. In such a machine one needs to characterise and minimise halo,and a the same time increase the beam quality, as one of the main de-sign goals. Needless to say that this is a very simplistic picture, and othervariables such as extracted energy, energy spread must also be consid-ered. In order to accomplish this task, usually a large number of designparameters, in the search space D (c.f. Figure 1), have to be considered.

In an ideal world you would run a number of high fidelity simulations(in some proportion to the size of D) to solve the problem. However, evenwith state-of-the art tools in cases of practical interest, it is impossible toaccomplish this task due to the prohibitive time to solution.

With the help of the surrogate model, there are 2 ways to tackle theproblem. The first is to solve the problem approximatively on a coarsersearch space (red grid in Figure 1) and then ”interpolate” to the true solu-tion from the cheap to run surrogate model. The second option is to usethe expensive high fidelity model to obtain u∗ ∈ D∗, but on a much smallerdomain (D∗). It is important to mention that the surrogate model does notreally reduce the search space. Rather, it is an approximation to the fullmodel over the area of the search space where one believes that the modelmatters the most. The goal of the surrogate model is to create a cheap tosample approximation of the full model.

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x∗

D∗ ⊂ D

D ⊂ Rd

Figure 1: Design parameter search space D, and one of the many pos-sible configurations x∗ of the accelerator, leading to a desirable solution(working point). The red grid is depicting the training points, from whichthe surrogate model will be constructed. The equidistance of these pointsis not necessary, however it is sufficient to introduce the overall concept.

2.2. Mathematical bases of UQWe briefly introduce the mathematical bases in the style and the nota-

tion of [19, 16, 20, 17, 14]. Let (Ω,F ,P) be a complete probability space,where Ω is the sample set and P is a probability measure on F , the σ−field(algebra) or Borel measure. Input uncertainties of the system has been dis-cretised and approximated by the random vector ξ = (ξ1, · · · , ξd) : Ω→ Rd,d ∈ N. The probability density function (pdf) of the random variable, ξk, isdenoted by ρ(ξk). Similarly, ρ(ξ) represents the joint pdf of ξ.

Let i be a multi-index i = (i1, . . . , id) ∈ Id,p and the set of multi-indicesId,p is defined by

Id,p = i = (i1, . . . , id) ∈ Nd0 : ‖i‖1 6 p, (1)

where ‖ · ‖1 is the l1 norm i.e., ‖ · ‖1 = i1 + · · ·+ id, and p is the polynomialorder.

All square integrable, second-order random processes with finite vari-ance output, u(ξ) ∈ L2 (Ω,F ,P), can be written as

u(ξ) =∞∑|i|=0

αiΨi(ξ). (2)

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Hence αi denotes the deterministic coefficients and Ψi(ξ) are the multi-variate PC basis functions [19, 10.1.1] [16]. Note that the uncertain QoI, u,is represented by a vector of deterministic parameters αi.

For the truncated PC Expansion (PCE) to order p in d dimensions of(2) we get

u(ξ) =∑i∈Id,p

αiΨi(ξ). (3)

The basis functions Ψi(ξ) in (3) are generated from

Ψi(ξ) =d∏

k=1

Ψik(ξk), i ∈ Id,p, (4)

where Ψik(ξk) are univariate polynomials of degree ik ∈ N0 := N ∪ 0,orthogonal with respect to ρ(ξk) (see, e.g., Table A.7), i.e.,

E[ΨikΨjk ] = 〈ΨikΨjk〉 =

∫Ψik(ξk)Ψjk(ξk)ρ(ξk)dξk = δikjkE[Ψ2

ik]. (5)

Here δikjk denotes the Kronecker delta and E[·] is the expectation operator.The number P of PC basis functions of total order P < p in dimension

d can be calculated to

P = |Id,p| =(p+ d)!

p!d!.

The PC basis functions Ψi(ξ) are orthogonal,

E[ΨiΨj ] = δi,jE[Ψ2i ], (6)

because of the orthogonality of Ψik(ξk) and the independence of ξk. Asp → ∞, the truncated PC expansion in (3) converges in the mean-squaresense, iff the following two conditions are fulfilled: 1) u(ξ) has finite vari-ance and 2) the coefficients αi are computed from the projection equation[20]

αi = E[u(·)Ψi(·)]/E[Ψ2i ]. (7)

2.3. Non-intrusive polynomial chaos expansionIn PC-based methods, one obtains the coefficients of the solution ex-

pansion either intrusively [29] or non-intrusively [30]. An intrusive approach

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requires significant modification of the deterministic solvers, which increasesthe number of equations by a factor P . As a consequence the intrusive PCexpansion method is P times more computationally expensive than a cor-responding non-intrusive model.

Non-intrusive methods on the other hand can make use of existing de-terministic solvers (M) as black boxes. First, one needs to generate aset of N deterministic or random samples of ξ, denoted by ξ(i)Ni=1. Thesecond step is to generate N realisations of the output QoI, u(ξ(i))Ni=1,with the available deterministic solver M and without any solver modifi-cations. The third and final step is to solve for the PC coefficients usingthe obtained realisations. Methods such as least squares regression [31],pseudo-spectral collocation [17], Monte Carlo sampling [32], and compres-sive sampling [33] are available. Along these lines an in depth discussionon least squares regression and compressive sampling can be found in[14, 3.1.1,3.1.2].

The mean, E[·], and variance, Var[·], of u(ξ) can be directly approxi-mated from the PC coefficients because of polynomial basis orthogonalitygiven by

E[u] = α0, (8)

andVar[u] =

∑i∈Id,pi 6=0

α2i E[Ψ2(ξi)]. (9)

A more complete description will be shown later in Section 2.5.

2.4. Global sensitivity analysisThe expensive, deterministic high fidelity particle accelerator model,

M, is described by a function u = M(x), where the input x is a pointinside D (c.f. Figure 1) and u is a vector of QoI’s. Finding correlations inthese high dimensional spaces is nontrivial, however it is vital for a deepunderstanding of the underling physics. For example, reducing the searchspace is of great interest in the modelling and optimisation process. Inthe spirit of Sobol’ [34], let u∗ = M(x∗) be the sought (true) solution.The local sensitivity of the solution u∗ with respect to xk is estimated by(∂u/∂xk)x=x∗. On the contrary, the global sensitivity approach does notspecify the input x = u∗, it only considers the model M(x). Therefore,global sensitivity analysis should be regarded as a tool for studying themathematical model rather than a specific solution (x = x∗).

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Following [34], the problems that can be studied, in our context, withglobal sensitivity analysis can be categorised the following way:

1. ranking of variables in u =M(x1, x2, . . . , xn)

2. identifying variables with low impact on u.

In this article, we use the Sobol’ indices [34] which are widely used due totheir generality. Results can be found in Section 4.5.

The first order PC-based Sobol’ index, Sk, represents the individualeffects of the random input ξk on the variability of u(ξ), and is given by

Sk =1

Var[u(ξ)]

∑i∈Ik

α2i E[Ψ2(ξi)], Ik = i ∈ Nd

0 : ik > 0, im6=k = 0. (10)

In order to compute Sk, all random inputs except ξk are fixed. As a conse-quence, Sk does not include effects arising from the interactions betweenξk and the other random inputs. This also means that Ik includes only thedimension k.

The fractional contribution to the total variability of u(ξ) due to parame-ter ξk, considering all other model parameters, is given by

STk =1

Var[u(ξ)]

∑i∈ITk

α2i E[Ψ2(ξi)] ITk = i ∈ Nd

0 : ik > 0. (11)

The set of multi indices ITk includes dimension k among others.Now we are in a position to rank the importance of the variables. The

smaller STk is, the less important the random input, ξk, becomes. We note,for the extreme case STk 1, the variable ξk is considered the be in-significant. In such a case, the variable can be replaced by its mean valuewithout considerable effects on the variability of u(ξ). We will make use ofthis fact when discussing the model problem and use STk as a measure toidentify the most important random inputs of the model.

If one is interested in the fraction of the variance that is due to the jointcontribution of the i-th and j-th input parameter, we can easily compute

Si,j =1

Var[u(ξ)]

∑i∈Ii,j

α2i E[Ψ2(ξi)] Ii,j = i ∈ Nd

0 : ik > 0. (12)

which describes this quantity.

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As an example to category 1 from above, consider a problem where xiand xj are two entries in the matrix of the second order moments of theinitial particle distribution within a simulation. We then find out that Si andSj are both much smaller than Si,j. Such a situation will indicate that otherentries in the matrix of second order moments significantly contribute. Forcategory 2, refer to [34, Section 7.], where an approximation of S is proven,when not considering all elements of x.

2.5. The UQTk based frameworkIn this section a detailed description is provided on how the particle

accelerator UQ framework is constructed. The framework is based on theUncertainty Quantification Toolkit (UQTk) [35], a lightweight C++/Pythonlibrary that helps performing basic UQ tasks including intrusive and non-intrusive forward propagation. UQTk can also be used for inverse mod-elling via Bayesian or optimisation techniques. The corresponding toolsused from UQTk are indicated in typewriter style in the following algorithm.

Let’s denote M as the black box solver, λ as the model parametersand y as the design or controllable parameter, with l equidistant values 1.The nonintrusive propagation of uncertainty from the d-dimensional modelparameter λ to the output ui = M(λ, yi) follows a collocation procedure,given a d-dimensional basis Ψ = (Ψ1, . . . ,Ψd) and K = (d+p)!

d!p!multivariate

basis terms with p being the maximal polynomial order.

Algorithm: generate for each yi (design or controllable), a PC surro-gate model

1. generate N = (p+ 1)d quadrature point-weight pairs (ξn, wn)(generate quad)

2. for each of quadrature point ξn compute corresponding model inputλn by

λn = λnj =K−1∑k=0

λjkΨk(ξn) j = 1, . . . , d. (13)

1For a fixed value of the design parameter, the surrogate construction algorithm isdescribed in [11].

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3. create the training points with high fidelity simulations (OPAL)

uni =M(λn, yi) i = 1, . . . , l. (14)

4. calculate the expectation via orthogonal projection (pce resp) usingquadrature

αki =〈uΨk〉〈Ψ2

k〉=

1

〈Ψ2k〉

N∑n=1

uni Ψk(ξn)wn, k = 0, . . . , K − 1. (15)

5. Given the computed αki values for each i and k, one assemblesthe PCE

ui =K−1∑k=0

αkiΨk(ξ), k = 0, . . . , K − 1. (16)

Remark 1: The input PC in Eq. (13) is assumed to be given by anexpert. For example, often only bounds for the inputs are known, in whichcase, Eq. (13) is simply a linear PC or just scaling from ξj ∈ [−1, 1] to λj ∈[aj, bj] for each j = 1, . . . , d. More explicitly stated, in Eq. (13) λj0 =

aj+bj2

,and λjk = δjk

bj−aj2

. Thus, Eq. (13) becomes

λnj =bj + aj

2+bj − aj

2ξnj . (17)

Remark 2: If samples ξn are randomly selected from the distribution ofξ, then the projection formula (15) still holds, as long as one sets wn = 1/Nfor all n, and it becomes an importance sampling Monte-Carlo.

2.5.1. Evaluation of the Surrogate modelHaving constructed the PC-coefficients, according to (15) the utility

pce eval can be used to evaluate ui (16).

2.5.2. Sensitivity AnalysisAs shown in Section 2.4, the same information used in the surrogate

model construction can be used in the sensitivity analysis. In the UQTkpce sens will compute the total and joint sensitivities along with the vari-ance fraction of each PC term individually.

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v-name l-bound u-boundv1 a1 b1v2 a2 b2...

......

vd ad bd

x = (x1, . . . , xl)

N Quadrature Points

Model evaluationsum =M(λm,x)m = 1 . . . lN

Using N samples

ui =K−1∑k=0

αkiΨk(ξ), i = 1 . . . l .

Surrogate Model ui Global Sensitivity Analysis

d Model Parameter λ

One design (or controllable) Parameter x

Figure 2: Uncertainty Quantification Framework

3. THE ACCELERATOR SIMULATION MODEL

For this discussion we briefly introduce OPAL-CYCL [36], one of thefour flavours of OPAL. OPAL will be used as the back-box solver denotedbyM in (14).

3.1. Governing EquationIn the cyclotron under consideration, the collision between particles can

be neglected because the typical bunch density is low. In time domain, thegeneral equations of motion for charged particles in electromagnetic fields

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can be expressed by

dp(t)

dt= q (cβ ×B + E) ,

wherem0, q, γ are the rest mass, the charge and the relativistic factor. Herewe denote p = m0cγβ as the momentum of a particle, c as the speed oflight, and β = (βx, βy, βz) as the normalised velocity vector. In general, thetime (t) and he position (x) dependent electric and magnetic vector fieldsare written in an abbreviated form as B and E.

If p is normalized by m0c, Eq. (18) can be written in Cartesian coordi-nates as

dpxdt

=q

m0cEx +

q

γm0

(pyBz − pzBy),

dpydt

=q

m0cEy +

q

γm0

(pzBx − pxBz), (18)

dpzdt

=q

m0cEz +

q

γm0

(pxBy − pyBx).

The evolution of the beam’s distribution function, f(x, cβ, t) : (IRM × IRM ×IR)→ IR, can be expressed by a collisionless Vlasov equation:

df

dt= ∂tf + cβ · ∇xf + q(E + cβ ×B) · ∇

cβf = 0. (19)

Here it is assumed that M particles are within the beam. In this particularcase, E and B include both externally applied fields and space chargefields.

E = Eext + Esc,

B = Bext + Bsc. (20)

All other fields are neglected.

3.2. Self FieldsThe space charge fields can be obtained by a quasi-static approxima-

tion. In this approach, the relative motion of the particles is non-relativisticin the beam rest frame, thus the self-induced magnetic field is practically

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absent and the electric field can be computed by solving Poisson’s equa-tion

∇2φ(x) = −ρ(x)

ε0, (21)

where φ and ρ are the electrostatic potential and the spatial charge densityin the beam rest frame. The electric field can then be calculated by

Esc = −∇φ, (22)

and back transformed to yield both the electric and the magnetic fields, inthe lab frame, as required in Eq. (20) by means of a Lorentz transformation.Because of the large vertical gap in our cyclotron, the contributions fromimage charges and currents are minor compared to space charge effects[37], and hence it is a good approximation to use open boundary condi-tions. Details on the space charge calculation methods utilised in OPALcan be found in [36, 38, 39]

3.3. External FieldsWith respect to the external magnetic field, two possible situations can

be considered. In the first situation, the real field map is available on themedian plane of the existing cyclotron machine using measurement equip-ment.

In most cases concerning cyclotrons, the vertical field, Bz, is measuredon the median plane (z = 0) only. Since the magnetic field outside themedian plane is required to compute trajectories with z 6= 0, the field needsto be expanded in the Z direction.

According to the approach given by Gordon and Taivassalo [40], byusing a magnetic potential and measured Bz on the median plane at thepoint (r, θ, z) in cylindrical polar coordinates, the 3rd order field can bewritten as

Bext(r, θ, z) = (z∂Bz

∂r− 1

6z3Cr,

z

r

∂Bz

∂θ− 1

6

z3

rCθ, Bz −

1

2z2Cz, (23)

where Bz ≡ Bz(r, θ, 0) and

Cr =∂3Bz

∂r3+

1

r

∂2Bz

∂r2− 1

r2∂Bz

∂r+

1

r2∂3Bz

∂r∂θ2− 2

1

r3∂2Bz

∂θ2,

Cθ =1

r

∂2Bz

∂r∂θ+

∂3Bz

∂r2∂θ+

1

r2∂3Bz

∂θ3, (24)

Cz =1

r

∂Bz

∂r+∂2Bz

∂r2+

1

r2∂2Bz

∂θ2.

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All the partial differential coefficients are computed on the median planedata by interpolation, using Lagrange’s 5-point formula.

In the second situation, a 3D field map for the region of interest is cal-culated numerically from a 3D model of the cyclotron. This is generallyperformed during the design phase of the cyclotron and utilises commer-cial software. In this case the calculated field will be more accurate, espe-cially at large distances from the median plane, i.e. a full 3D field map canbe calculated. For all calculations in this paper, we use the Gordon andTaivassalo [40] method.

For the radio frequency cavities, a radial voltage profile V (r) along theradius of the cavity is used. The gap-width, g, is included in order to correctfor the transit time. For the time dependent field,

∆Erf =sin τ

τ∆V (r) cos(ωrft− φ), (25)

with F denoting the transit time factor (F = 12ωrf∆t), and ∆t the transit time

defined by∆t =

g

βc. (26)

In addition, a voltage profile varying along radius will give a phase com-pression of the bunch, which is induced by an additional magnetic fieldcomponent Bz in the gap,

Bz '1

gωrf

dV (r)

drsin(ωrft− φ). (27)

From (27) we can calculate a horizontal deflection, α, as

α ' q

m0βγcωrft

dV (r)

drsin(ωrft− φ). (28)

Finally, in this paper, both the external fields and space charge fieldsare used to track particles for one time step using a 4th order Runge-Kutta(RK) integrator. This means the fields are evaluated for four times in eachtime step. Space charge fields are assumed to be constant during one timestep because their variation is typically much slower than that of externalfields.

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Figure 3: The cyclotron model problem setup. The two red lines indicatingthe 2 double gap flat-top resonators, the blue line represents a collimator,and the yellow circle stands for the initial conditions.

4. APPLICATION OF THE UQ FRAMEWORK

In order to demonstrate the usefulness and strength of UQ, consider asimplified model of the PSI Injector 2 cyclotron which is sketched in Fig-ure 3. The simplifications are as follows: 1) only energies up to 8.5 MeV(turn 10) are considered to reduce the computational burden. 2) a Gaus-sian distribution, linearly matched to the injection energy of 870 keV, isused for the initial conditions. 3) the magnetic field and RF structures arethe same as in our full production simulation. 4) Pr andR are obtained fromequilibrium orbit simulations, and 5) one collimator is introduced in orderto mimic bunch shaping. Full scale high fidelity simulations of this kind canbe found in [41, 42], where similar physics goals have being pursued.

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4.1. Model parametersIn typical design studies of high power cyclotrons, the high number of

model parameters are such that one can not fully scan their entire range.For this feasibility study, one model parameter out of a family of three im-portant categories (c.f. Figure 3) was choosen:

1. initial conditions: model parameter 〈xpx〉, correlation between initialthe x and px phase space variables

2. collimator settings: model parameter ∆C1 position of the collimator3. rf phase settings: model parameter φ1 defines the phase of the ac-

celeration cavity.

From previous experience, these three categories have the most influencewhen designing and optimising high precision models of high power cy-clotrons. The relationship of the parameters with uncertainties, λ1, λ2, λ3,is shown in Figure 2.

4.2. Quantities of interest (QoI)The phase space spanned by M macro particles, in the high fidelity

OPAL model (simulation), is given by (qi(t),pi(t)) ∈ Γ ⊂ IR(2M+1) andi = x, y, z. We identify a subset of interesting QoI’s such as:

1. εx =√〈q2xp2x〉 − 〈qxpx〉2 the rms projected emittance and x the rms

beam size,2. the kinetic energy E and rms energy spread ∆E,3. ht =

〈q4x〉

〈q2x〉2− c, the halo parameter in x-direction at end of turn t with

c ∈ IR, a distribution dependent normalisation constant.

The rms beam size x is one of the better quantities that can be di-rectly measured and hence among the first candidate for characterisationof the particle beam. A measure of the projected phase space volume isthe emittance εx. This quantity is often used for the estimation of the beamquality. The two energy related parameters E and ∆E are target valuesto achieve, the first one closely related to the experiment where the parti-cle beam is designed for, the energy spread ∆E is directly related to thebeam quality in the case of the presented model problem. Minimizing thehalo of the particle beam is equal to minimizing losses, the most importantquantity to optimize in high power hadron accelerators. In the formulationof ht, this parameter is deviating from 1 iff the initial chooses distribution

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is changing. If the initial distribution is a stationary distribution, this mea-sure can be attributed to the mechanism of halo generation, in case of adeviation from the value 1.

In the case of a high intensity cyclotron model, we choose the control-lable parameter y as the average current.

4.3. UQ model setupThe controllable parameters are not modelled with polynomials, but

rather given by 10 equidistant values from 1 . . . 10 mA. As a next step,the polynomial type for the model parameter is chosen according to theWiener-Askey scheme (cf. Appendix Appendix A). The distribution of thethree model parameters 〈xpx〉, ∆C1, and the phase φ1, are modelled ac-cording to an uniform distribution using polynomials of the Legendre type.The bounds of the distribution are given in Table 1. Other parameters for

Table 1: Upper and lower bounds of the design parameters

v-name l-bound u-bound〈xpx〉 -0.5 0.5∆C1 (mm) 0 5φ1() -20 20

the UQ model are listed in Table 5.

Table 2: Summary of UQ related parameters for the presented results.The dimension for all the experiments are d = 3. The one controllableparameter y has length l = 10.

Parameter Meaning Experiment 3 2 1p order of surrogate construction 2 3 4

quadrature points per dimension (p+ 1) 3 4 5N quadrature points N = (p+ 1)d 27 64 125K polynomial basis terms K = (d+ p)!/d!p! 10 20 35N · l number of high-fidelity runs 270 640 1250

17

4.4. High Fidelity Simulations vs. Surrogate ModelAs a first method to determine the validity of the surrogate model, the

values of the high fidelity OPAL simulations on the x-axis and the valuesof the surrogate model on the y-axis were compared. The distance of thecorresponding point to the line x = y, is a measure of surrogate model’squality. The QoI’s, as defined in Section 4.2, are compared for a subsetof controllable parameters: 1, 5, 8, and 10 mA, and for 3 different orders ofthe surrogate model, as described in Table 5. All data from the surrogatemodel and the high fidelity model are taken at the end of turn 10 in ourmodel problem.

Overall the expected convergence is observed when increasing p asshown in Figure 4 through Figure 9, and furthermore this is supported bythe L2 error norm shown in Section 4.6.

4.4.1. Projected Emittance & Beam sizeGiven the fact that the emittance is a very sensitive quantity, measuring

phase space volume, it is surprising, but also promising, that such a goodagreement between the surrogate model and the high fidelity model canbe achieved. This is graphically illustrated in Figure 4 and Figure 5. Themaximum error in % is given in Table 3 and Table 4, and is below 7% forall considered cases.

Table 3: Maximum error in % between the high fidelity and surrogate modelfor the projected emittance εx of the beam.

P = 4 P = 3 P = 2

I = 1 (mA) 1.94 2.81 3.35I = 5 (mA) 5.04 4.77 2.79I = 8 (mA) 4.89 4.95 6.70I = 10 (mA) 3.6 2.78 5.60

18

Table 4: Maximum error in % between the high fidelity and surrogate modelfor the rms beam size x of the beam.

P = 4 P = 3 P = 2

I = 1 (mA) 0.70 0.87 1.03I = 5 (mA) 2.32 2.90 3.49I = 8 (mA) 1.04 3.33 1.86I = 10 (mA) 1.33 1.98 1.39

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

High Fidelity OPAL Simulation

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Poly

nom

ial Surr

ogate

Experiment 3 εx(mm−mr)Experiment 2 εx(mm−mr)Experiment 1 εx(mm−mr)y=x

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

High Fidelity OPAL Simulation

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Poly

nom

ial Surr

ogate

Experiment 3 εx(mm−mr)Experiment 2 εx(mm−mr)Experiment 1 εx(mm−mr)y=x

2 3 4 5 6 7 8

High Fidelity OPAL Simulation

2

3

4

5

6

7

8

Poly

nom

ial Surr

ogate

Experiment 3 εx(mm−mr)Experiment 2 εx(mm−mr)Experiment 1 εx(mm−mr)y=x

3 4 5 6 7 8 9

High Fidelity OPAL Simulation

3

4

5

6

7

8

9

Poly

nom

ial Surr

ogate

Experiment 3 εx(mm−mr)Experiment 2 εx(mm−mr)Experiment 1 εx(mm−mr)y=x

I = 1 mA I = 5 mA

I = 8 mA I = 10 mA

Figure 4: Projected emittance εx (mm-mr) for all 3 experiments describedin Table 5.

19

5 6 7 8 9 10 11 12

High Fidelity OPAL Simulation

5

6

7

8

9

10

11

12

Poly

nom

ial Surr

ogate

Experiment 3 xrms(m)

Experiment 2 xrms(m)

Experiment 1 xrms(m)

y=x

6 7 8 9 10 11 12 13

High Fidelity OPAL Simulation

6

7

8

9

10

11

12

13

Poly

nom

ial Surr

ogate

Experiment 3 xrms(m)

Experiment 2 xrms(m)

Experiment 1 xrms(m)

y=x

7 8 9 10 11 12 13 14 15

High Fidelity OPAL Simulation

7

8

9

10

11

12

13

14

15

Poly

nom

ial Surr

ogate

Experiment 3 xrms(m)

Experiment 2 xrms(m)

Experiment 1 xrms(m)

y=x

7 8 9 10 11 12 13 14 15 16

High Fidelity OPAL Simulation

7

8

9

10

11

12

13

14

15

16

Poly

nom

ial Surr

ogate

Experiment 3 xrms(m)

Experiment 2 xrms(m)

Experiment 1 xrms(m)

y=x

I = 1 mA I = 5 mA

I = 8 mA I = 10 mA

Figure 5: RMS beam size x (mm) for all 3 experiments described in Ta-ble 5.

20

4.4.2. Final EnergyThe energy dependence shown in Figure 6 for 10 mA, illustrated the

same expected behaviour for all other intensities. This is because of thesmall gain the third harmonic cavity is suppose to deliver (in the PSI In-jector 2 we use the third harmonic cavity for acceleration). For the givenexperiment only the last two turns are contributing. This fact is even betterillustrated, when looking at the maximum error which is ≤ 0.07 %, as seenin Table 5.

8.4 8.5 8.6 8.7 8.8High Fidelity OPAL Simulation

8.4

8.5

8.6

8.7

8.8

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

I = 10 mA

Figure 6: Final Energy E (MeV) for I = 10 mA, and all experiments de-scribed in Table 5.

Table 5: Maximum error in % between the high fidelity and surrogate modelfor the final energy of the beam.

P = 4 P = 3 P = 2

I = 1 (mA) 0.013 0.017 0.070I = 5 (mA) 0.013 0.036 0.066I = 8 (mA) 0.014 0.029 0.057I = 10 (mA) 0.010 0.027 0.056

21

4.4.3. RMS Energy SpreadDespite the fact the rms energy spread is influenced by space charge,

the collimation, and the change in phase, a very good agreement withabsolute deviations ≤ 5% where obtained. Table 6 shows details.

21 22 23 24 25 26 27 28 29High Fidelity OPAL Simulation

21

22

23

24

25

26

27

28

29

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

15 20 25 30 35 40 45High Fidelity OPAL Simulation

15

20

25

30

35

40

45

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

25 30 35 40 45 50High Fidelity OPAL Simulation

25

30

35

40

45

50

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

30 35 40 45 50 55High Fidelity OPAL Simulation

30

35

40

45

50

55

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

I = 1 mA I = 5 mA

I = 8 mA I = 10 mA

Figure 7: Energy spread ∆E (keV) for all 3 experiments described in Ta-ble 5.

22

Table 6: Maximum error in % between the high fidelity and surrogate modelfor the energy spred ∆E of the beam.

P = 4 P = 3 P = 2

I = 1 (mA) 0.97 1.67 1.62I = 5 (mA) 2.56 1.04 1.29I = 8 (mA) 2.56 2.75 4.65I = 10 (mA) 3.00 3.70 4.48

23

4.4.4. Halo ParametersThe halo parameter was evaluated at turn 5 (Figure 8) and at turn 10

(Figure 9). As anticipated the halo grows and the surrogate model has amaximum absolute error of ≤ 5%, again a very good accuracy.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8High Fidelity OPAL Simulation

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0High Fidelity OPAL Simulation

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

0.1 0.2 0.3 0.4 0.5 0.6 0.7High Fidelity OPAL Simulation

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

0.1 0.2 0.3 0.4 0.5 0.6High Fidelity OPAL Simulation

0.1

0.2

0.3

0.4

0.5

0.6

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

I = 1 mA I = 5 mA

I = 8 mA I = 10 mA

Figure 8: The dimensionless halo parameter h after turn 5 for all 3 experi-ments described in Table 5.

24

0.0 0.5 1.0 1.5 2.0 2.5High Fidelity OPAL Simulation

0.0

0.5

1.0

1.5

2.0

2.5

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4High Fidelity OPAL Simulation

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

0.2 0.4 0.6 0.8 1.0High Fidelity OPAL Simulation

0.2

0.4

0.6

0.8

1.0

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0High Fidelity OPAL Simulation

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Poly

nom

ial Surr

ogate

P=4

P=3

P=2

y=x

I = 1 mA I = 5 mA

I = 8 mA I = 10 mA

Figure 9: The dimensionless halo parameter h after turn 10 for all 3 exper-iments described in Table 5.

25

4.5. Sensitivity AnalysisSk in (10) can be interpreted as the fraction of the variance in modelM

that can be attributed to the i-th input parameter only. STk in (11) measuresthe fractional contribution to the total variance due to the i-th parameterand its interactions with all other model parameters. In the sequel an anal-ysis based of STk is shown for the model problem.

Figure 10 showing, for a subset of the controllable parameter I, sensi-tivities of the QoI’s with respect to the model parameters. The ploynomialorder is P = 4, the similar correlations for other orders are not shown.Correlations, for example the insensitivity of the energy, and x, px or the

Ekin(MeV) εx(mm−mr) ∆E(keV) h5 h100.0

0.2

0.4

0.6

0.8

1.0

Sensi

tivit

y

xpx C1 (mm) ∆φrf( )

Ekin(MeV) εx(mm−mr) ∆E(keV) h5 h100.0

0.2

0.4

0.6

0.8

1.0

Sensi

tivit

y

xpx C1 (mm) ∆φrf( )

Ekin(MeV) εx(mm−mr) ∆E(keV) h5 h100.0

0.2

0.4

0.6

0.8

1.0

Sensi

tivit

y

xpx C1 (mm) ∆φrf( )

Ekin(MeV) εx(mm−mr) ∆E(keV) h5 h100.0

0.2

0.4

0.6

0.8

1.0

Sensi

tivit

y

xpx C1 (mm) ∆φrf( )

I = 1 mA I = 5 mA

I = 8 mA I = 10 mA

Figure 10: Experiment 1: Global sensitivity analysis for intensities of 1,5,8and 10 mA

significant energy phase correlation, are consistent with what is antici-pated. A very mild dependency on x, px is observed and expected. Thereis a phase correlation appearing in the case of I = 5 mA, which seems

26

to be suppressed at other intensities, and the initial correlation of the dis-tribution seams to become insignificant. A closer inspection of the phasespace, beyond the scope of that paper, hints that the halo at this intensityhas a minimum. This could explain the observed behaviour and is subjectto a deeper investigation.

These are very interesting findings that can guide new designs butalso improve existing accelerators, and shows quintessential the merit andpower of such a sensitivity analysis.

4.6. Error Propagation and L2 ErrorIn Figure 11, the L2 error

L2 =||u− u||2||u||2

between the surrogate model u and u, the high fidelity OPAL model, isshown for E, the final energy of the particle beam and all values of thecontrollable parameter I. The mean value and variance are shown on theleft y-axis. We can now precisely define the error and the dependency ofthe surrogate model on P . The expected convergence of the surrogatemodel as function of P is shown for one model parameter only, because ofthe similar behaviour in the other considered parameters. This clearly helpin choosing an appropriate order of the surrogate model.

4.7. ExtrapolationThe surrogate model is constructed by selecting an appropriate num-

ber of training points in order to sample the input uncertainties of the de-sign parameter space. These finite number of training points are depictedas yellow points in Figure 12. However, with the surrogate model we canchoose any point within the lower and upper bound specified (ai, bi in (17))in order to obtain λ in (13). This we call extrapolation. In Figure 12 thered points are arbitrarily chosen, within the specified bounds and they arevery we within the bounds of the surrogate model and the 95% confidencelevel (CL) obtained by evaluating the Student-t test.

4.8. PerformanceThe presented surrogate model is the most simple, but gives for the non

trivial model problem, statistically sound results. This fact and the remark

27

8.56

8.565

8.57

8.575

8.58

8.585

8.59

8.595

8.6

8.605

8.61

0 2 4 6 8 10

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

E (

MeV

)

L2

erro

r

I (mA)

Experiment 1 L2, P=4Experiment 2 L2, P=3Experiment 3 L2, P=2

Experiment 1 E (MeV)Experiment 2 E (MeV)Experiment 3 E (MeV)

Figure 11: Medium values, and variances are shown on the left y-axis forthe extraction energy E. The global L2 error between the high fidelity andthe surrogate model, for the final energy of the particle beam, is shown onthe right y-axis.

that the evaluation of the surrogate model is ∼ 800× faster than the high fi-delity model (400 seconds v.s. 0.5 seconds) opens up unprecedented pos-sibilities in research areas such as on-line modelling and multi-objective[43, 44] optimization of charged particle accelerators.

5. CONCLUSIONS

A sampling-based UQ approach is presented to study, for the first time,the effects of input uncertainties on the performance of particle acceler-ators. A particular, but complex, example in the form of a high intensitycyclotron was used to demonstrate the usefulness of the surrogate modelas well as the global sensitivity analysis via computing the total Sobol’ in-dices. The presented physics problem is a model problem, with the aimof demonstrating the usefulness and applicability of the presented UQ ap-

28

ξ0 20 40 60 80 100 120

(m

m)

x~

8

9

10

11

12

13

14

15Surrogate Model, P=4CL95%Training pointsExtrapolation

Figure 12: The surrogate mode for x, together with training and extrapola-tion points. The 95% CL of the model is also shown.

proach. However, we claim to present a problem that can be recognisedas a template for many high intensity modelling attempts, and beyond.

The proposed UQ approach is based on polynomial chaos expansionusing the UQTk software. The goal is to achieve an accurate estimation ofsolution statistics using a minimal number of high fidelity simulations. Forseveral QoI’s a surrogate model was constructed, the validity is proofed bycomparing to a high fidelity model. L2 error norms showing the expectedconvergence in regard to the degree of the polynomial chaos expansion.For the rms beam size (x), extrapolation points, i.e. points that are notused in the training set, where evaluated and compared to the statisticalexpectations from the model. We found that the values are consistent with

29

the surrogate model and very well within the 95% CL.The Sobol’ based global sensitivity analysis was in line with the expec-

tation from the physics evaluation of the model problem.A tremendous speedup of 800× was observed, comparing the time to

solution of the surrogate model to the high fidelity model. This opens uppossibilities for on-line modelling and multi-objective optimization of com-plex particle accelerators.

Future research includes the application to real word problems in thearea of high intensity hadron machines, i.e. performance enhancementsof exiting machines and design improvements of future machines [41]. Inthe area of proton therapy, we focus on understanding the uncertainty ofaccelerator parameters, in relation to the applied radiation dose to the pa-tient.

In this paper, conceptional we followed the simplest approach towardsUQ. Given the encouraging results, we plan to enhance this model byusing Hermite chaos, and going to higher dimensions, which implies theuse of sparse methods or latin hyper cube sampling.

Particle accelerators in general create a vast amount of high qualitydata, including the QoI’s we have considered. Including such data into thethe model, or solving an inverse problem could be interesting researchtopics for the future.

6. ACKNOWLEDGMENTS

Dr. N. Pouge for critical comments and english proof reading. Ms. V.Rizzioglio for many fruitful discussion and help with the art work. Dr. K.Sargsyan & Dr. B. Debusschere for UQTk and UQ related discussions.

Appendix A. Wiener-Askey PC

30

Table A.7: The correspondence of Wiener-Askey PC and the pdf of therandom variables [20].

ρ(ξk) Polynomial SupportBeta Jacobi [a,b]

Uniform Legendre [a,b]Gaussian Hermite (-∞,+∞)Gamma Laguerre (0,+∞)

31

Appendix B. Legendre polynomials

The Legendre polynomials, or Legendre functions of the first kind (B.1),(Whittaker and Watson 1990, p. 302), are solutions to the Legendre differ-ential equation, a second-order ordinary differential equation

(1− x2)d2y

dx2− 2x

dy

dx+ l(l + 1)y = 0. (B.1)

In case of l ∈ N , the solutions are polynomials Pn. The first few polynomi-als relevant to this paper are shown in (B.2).

P0(x) = 1

P1(x) = x

P2(x) = 1/2(3x2 − 1) (B.2)P3(x) = 1/2(5x3 − 3x)

P4(x) = 1/8(35x4 − 30x2 + 3)

. . .

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